References for gradient systems

Question

I am interested in the gradient system

$$\dot{x}(t)=-\nabla f(x(t))$$

where $f:\mathbb{R}^n \to \mathbb{R}$ is a $C^{1,1}$ function (that is, a differentiable function whose gradient is Lipschitz continuous). I would be grateful if someone could propose some reference books related to this system.

Answer 1

If $f$ is convex, then the gradient flow $\dot{x} = - \nabla f (x)$ is doing continuous-time gradient descent. A book on gradient flows is the following one:

• Uwe Helmke, John B. Moore, Optimization and Dynamical Systems, 2nd edition, March 1996.